Abstract: The weight functions for which , the Riemann-Liouville fractional integral operator of order , is bounded from to , , , are characterized. Further, given , with , the weight functions a.e. (resp. a.e.) for which there is a.e. (resp. a.e.) so that is bounded from to are characterized. Analogous results are obtained for the Weyl fractional integral. The method involves the use of complex interpolation of analytic families of operators to obtain similar results for fractional "one-sided" maximal function operators which are of independent interest.